A line segment is bisected by a line with the equation 4 y + 3 x = 8 . If one end of the line segment is at ( 1 , 8 ), where is the other end?

1 Answer
Cesareo R.
May 16, 2016

Over the straight line 4y+3x+19=0

Explanation:

The straight 4y+3x-8 = 3x +4(y-2)=0 passes by point (0,2) and has the direction given by the vector v = (4,-3). In parametric form can be written as
p = p_0+lambda v with p_0=(0,2)
Given a generic straight point p, the symmetrical point to q = (1,8) regarding the straight line, is given by
q_S = q +2(p-q) = 2p-q = 2p_0-q+2lambda v which is a straight line parallel to the initial straight whose equation is given by
q_S=2(0,2)-(1,8)+2\lambda(4,-3)
In Cartesian coordinates we have
x=-1+8\lambda
y = 4-8-6\lambda
The Cartesian representation gives us
4y+3x+19=0

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